Question

Diagonals of a trapezium PQRS intersect each other at the point O, PQ || RS and PQ = 3 RS. Find the ratio of the areas of triangles POQ and ROS.

Answer 1

Given PQRS is a trapezium in which PQ || RS and PQ = 3 RS

⇒ `("PQ")/("RS") = 3/1`   ...(i)

In ∆POQ and ∆ROS,

∠SOR = ∠QOP  ...[Vertically opposite angles]

∠SRP = ∠RPQ   ...[Alternate angles]

∴ ∆POQ ~ ∆ROS  ...[By AAA similarity criterion]

By property of area of similar triangle,

`("ar(∆POQ)")/("ar(∆SOR)") = ("PQ")^2/("RS")^2`

= `("PQ"/"RS")^2`

= `(3/1)^2`   ...[From equation (i)]

⇒ `("ar(∆POQ)")/("ar(∆SOR)") = 9/1`

Hence, the required ratio is 9 : 1.